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By Boris Botvinnik

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Example text

Fundamental group of a finite CW -complex. Here we prove a general result showing how to compute the fundamental group π1 (X) for arbitrary CW -complex X . Remark. Let X be a path-connected. If a map S 1 −→ X sends a base point s0 to a base point x0 then it determines an element of π1 (X, x0 ); if f sends s0 somethere else, then it defines an element of the group π(X, f (s0 )), which is isomorphic to π1 (X, x0 ) with an isomorphism α# . The images of the element [f ] ∈ π(X, f (s0 )) in the group π1 (X, x0 ) under all possible isomorphisms α# define a class of conjugated elements.

2. 2 implies that e(σ1 , . . , σk ) is homeomorphic to an open cell of dimension d(σ) = (σ1 − 1) + (σ2 − 2) + · · · + (σk − k). Remark. Let (v1 , . . , vk ) ∈ E(σ) \ E(σ), then the k -plane π = v1 , . . , vk does not σj belong to e(σ). Indeed, it means that at least one vector vj ∈ Rσj −1 = ∂ H . Thus dim(Rσj −1 ∩ π) ≥ j , hence π ∈ / e(σ). 3. A collection of k n cells e(σ) gives G(n, k) a cell-decomposition. Proof. We should show that any point x of the boundary of the cell e(σ) belongs to some cell e(τ ) of dimension less than d(σ).

Now let f : X −→ Y be a map; it induces a homomorphism f∗ : πn (X) −→ πn (Y ). 5 There is a special name for the group π1 (X) : the fundamental group of X . 2. Prove that if f, g : X −→ Y are homotopic maps of pointed spaces, than the homomorphisms f∗ , g∗ : πn (X) −→ πn (Y ) coincide. 3. Prove that πn (X × Y ) ∼ = πn (X) × πn (Y ) for any spaces X, Y . 2. One more definition of the fundamental group. The definition above was two general, we repeat it in more suitable terms again. e. such maps ϕ : I −→ X that ϕ(0) = ϕ(1) = x0 .

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Algebraic Topology Notes(2010 version,complete,175 pages) by Boris Botvinnik


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Categories: Geometry And Topology